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I haven’t posted a new proof in a while, and that will probably continue for a few more weeks. I currently have a bandaged up left hand due to a kitchen knife accident, which is making it slower for me to work on the computer. And I also have my wife working from home due to COVID-19 travel restrictions, which breaks up my days with many more distractions than normal, making it very hard to concentrate in the long blocks of time necessary to research and write up a new proof.

Rest assured that the site is not abandoned! I’ll be back with more new proofs, hopefully within a few weeks.

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A waveguide is a structure that restricts the motion of waves, disallowing propagation in certain directions, and thus concentrating the energy of the wave to propagate in specific other directions. An example of a waveguide is an optical fibre, which is basically a long, thin string of flexible glass or transparent polymer. Light entering one end is channelled along the fibre, unable to escape from the sides, and emerges at almost the same brightness from the far end.

Normally light and other electromagnetic waves, as well as other waves such as sound, spread out in three dimensions. As the energy spreads out to cover more space, conservation of energy causes the wave amplitude to fall off according to the inverse square law: wave amplitude falls as the reciprocal of the square of the distance from the source.

With a waveguide, propagation of the wave can be restricted to a single dimension so the energy doesn’t spread out, resulting in all of the energy being transmitted to the far end (minus a small fraction that may be absorbed or otherwise lost along the way). Sound waves, for example, can be guided by simple hollow tubes, the sound preferring to propagate along the interior air channel than penetrate the tube walls. This is the principle behind medical stethoscopes and old fashioned speaking tube systems.

Another type of waveguide is a transmission line, which is a pair of electrical cables used to transmit alternating current (AC) electrical power. The cables can simply be parallel wires in close proximity, or a coaxial cable, in which an insulated wire runs down the core of tubular conductor. Domestic AC power has a frequency of 50 to 60 hertz, which is low compared to the kilohertz range of radio frequencies. Transmission lines can carry electromagnetic waves up to frequencies of around 30 kHz. Above this, paired wires start to radiate radio waves, so they become inefficient and a different type of waveguide is used.

Radio waveguides are commonly hollow metal tubes. Radio waves travel along the tube, and the conductive metal prevents the waves from leaking to the outside. Such waveguides are used to transmit radio power in radar systems and microwaves in microwave ovens. Anywhere there is a cavity bounded by regions that waves cannot pass through, a waveguide effect can be generated.

Radio waves travel easily through the Earth’s atmosphere, to and from transmission towers and the various wireless devices we use. However the bulk of the Earth is opaque to radio waves; you generally need a mostly unobstructed line of sight, barring relatively thin obstructions like walls.

But there is another region of the Earth that is opaque to (at least some) radio waves. The ionosphere is the region of the atmosphere in which incoming solar radiation ionises the atmospheric gases (mentioned previously in 31. Earth’s atmosphere). It lies between approximately 60 to 1000 km altitude. Since ionised gas conducts electricity, low frequency radio waves cannot pass through it (higher frequencies oscillate too rapidly for the ionised particles to respond).

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Opacity of the Earth’s atmosphere as a function of electromagnetic wavelength. Long wavelength (low frequency) radio waves are blocked by the ionosphere (right). Other parts of the electromagnetic spectrum are blocked by other aspects of the atmosphere. (Modified from a public domain image by NASA, from Wikimedia Commons.)

Radio waves with wavelengths longer than about 30 metres—or frequencies below about 10 MHz—are thus trapped in the atmosphere between the Earth’s surface and the ionosphere. This forms a waveguide which can carry so-called shortwave radio signals around the world, alternately bouncing off the ionosphere and the Earth’s surface.

There are also natural sources of low frequency radio waves. Lightning flashes in storm systems produce huge discharges of electrical energy, and the sudden release of this energy generates radio waves. If you’ve ever listened to a radio during a thunderstorm you’ll be familiar with the bursts of static caused by strokes of lightning. Lightning generates broadband radio emissions, meaning it covers a wide range of radio frequencies, including the very low frequencies that are guided by the ionospheric waveguide.

Atmospheric scientists measure the amount of lightning around the world by monitoring tiny changes in the Earth’s magnetic field, of the order of picoteslas, caused as these radio waves pass by. The sensitive detectors they use can detect lighting strikes anywhere on the planet. There are a few specific radio frequencies at which the lightning strikes turn out to be especially strong. The following plot shows the intensity of magnetic field fluctuations as a function of radio frequency.

Measurements of magnetic field fluctuation amplitude versus radio wave frequency, averaged over a year of observation, at Maitri Research Station, Antarctica. (Figure reproduced from [1].)

The first peak in the observed radio spectrum is at 7.8 Hz, followed by peaks at 14.3 Hz, 20.8 Hz, and roughly every 6.5 Hz thereafter. People familiar with wave theory will recognise from the pattern that these are likely resonance frequencies, with a fundamental mode at 7.8 Hz, followed by overtones. A wave resonance occurs when an exact number of wavelengths fits into a confined cavity. The wave propagates and bounces around and, because of the precise match with the cavity size, reflected waves end up with peaks and troughs in the same physical position, reinforcing one another. So at the specific resonance frequency, the wave builds up in intensity, while at other frequencies the waves self-interfere and rapidly die down. These resonance frequencies, which are measured at many research stations around the world, are known as Schumann resonances.

The Irish physicist George Francis FitzGerald first anticipated the existence of Schumann resonances in 1893, but his work was not widely circulated. Around 1950, the German physicist Winfried Otto Schumann performed the theoretical calculations that predicted the resonances may be observable, and made efforts to observe them. But it was not until 1960 that Balser and Wagner made the first successful observations and measurements of Schumann resonances.[2]

What causes the radio waves produced by lightning flashes to have a resonance at 7.8 Hz? Well, radio waves travel at the speed of light, so let’s divide the speed of light by 7.8 to see what the wavelength is: the answer is 38,460 km. If you’ve been paying attention to many of these articles, you’ll realise that this is very close to the circumference of the Earth.

Radio waves with a frequency of 7.8 Hz are travelling around the world in the waveguide formed by the Earth and the ionosphere, and returning one wavelength later to constructively interfere and reinforce themselves, producing a measurable peak in Earth’s magnetic field fluctuations at 7.8 Hz. The resonance peak is broad and a little different to 7.5 Hz (the speed of light divided by the circumference of the Earth) because the geometry of a spherical cavity is more complicated than a simple circular loop – effectively some propagation paths are shorter because the waves don’t all take a great circle route.

Illustration of Schumann resonances in the Earth’s atmosphere. The ionosphere keeps low frequency radio waves confined to a channel between it and the Earth. Waves propagate around the Earth. At specific frequencies the peaks and troughs line up, producing a resonance that reinforces those frequencies. The blue wave fits six wavelengths around the Earth, the red wave fits three. The fundamental frequency Schuman resonance of 7.8 Hz fits one wave. Not to scale: the ionosphere is much closer to the surface in reality. (Public domain image by NASA/Simoes.)

So Schumann resonances are an observed phenomenon that has a natural explanation – if the Earth is a globe.

If the Earth were flat, then any ionosphere above it would be flat as well, and would still form a waveguide for low frequency radio waves. However it would not be a closed waveguide. Radio waves would propagate out the edges and be lost to space, meaning there would be no observable magnetic field resonances at all. And even if there were an opaque radio wall of some sort at the edge of the flat Earth, the size and geometry of the resulting cavity would be different, resulting in a different set of resonance frequencies, more akin to the frequencies of a vibrating disc, which are not evenly spaced like the observed Schumann resonances.

And so Schumann resonances provide another proof that the Earth is a globe.

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The French physicist Henri Becquerel discovered the phenomenon of radioactivity in 1896, while performing experiments on phosphorescence – the unrelated phenomenon that causes “glow in the dark” materials to glow for several minutes after being exposed to light. He was interested to see if phosphorescence was related to x-rays, discovered only a few months earlier by Wilhelm Roentgen. In his experiments, Becquerel noticed that uranium salts could darken photographic film, even if wrapped in black paper so that no light could fall on the film, and even from non-phosphorescent uranium samples. The conclusion was that some sort of penetrating rays were being emitted by the uranium itself, without being excited by external energy.

Marie and Pierre Curie quickly discovered other radioactive elements, and Becquerel himself discovered by experimenting with magnets that there were three different types of radioactive radiation: two deflected in different directions by a magnetic field and one not deflected at all. In 1899, Ernest Rutherford characterised the first two types, naming them alpha and beta particles, with positive and negative electric charges. Becquerel measured the mass/charge ratio of beta particles in 1900 and determined that they were the same as the electrons discovered by J. J. Thomson in 1897. In 1907 Rutherford showed that alpha particles were the nuclei of helium atoms. And in 1914, he showed that the third type of radiation, named gamma rays, were a form of electromagnetic radiation.

This was an exciting time in physics, and our understanding of atomic structure was revolutionised within the space of two decades. Besides discovering the basic structure of the atom and how it related to the phenomenon of radioactive decay, several peripheral phenomena also came to the attention of scientists.

One observation was that atoms in the atmosphere were sometimes ionised, or “electrified” as the scientists of the time described it. Ionisation is the process of electrons being stripped off neutral atoms, to form negatively charged free electrons and positively charged atomic ions (consisting of the atomic nucleus and a less-than-full complement of electrons). It was clear that radioactive rays could ionise atoms in the air, and so scientists assumed that it was radiation from radioactive elements in the ground that was ionising the air near ground level.

Except strangely the amount of ionising radiation in the atmosphere seemed to increase with increasing altitude. German physicist and Jesuit priest Theodor Wulf invented in 1909 a portable electroscope capable of measuring the ionisation of the atmosphere. He used it to investigate the source of the ionising radiation by measuring ionisation at the base and the top of the Eiffel Tower. He found that the ionisation at the top of the 300 metre tower was a bit over half that at ground level, which was higher than he expected, since theoretically he expected the ionisation to drop by half every 80 metres, so to be less than one tenth the ionisation at ground level. He concluded that there must be some other source of ionising radiation coming from above the atmosphere. However, his published paper was largely ignored.

In 1911, the Italian physicist Domenico Pacini measured the ionisation rates in various places, including mountains, lakes, seas, and underwater. He showed that the rate dropped significantly underwater, and concluded that the main source of radiation could not be the Earth itself. Then in 1912, Austrian physicist Victor Hess took some Wulf electroscopes up in a hot air balloon to altitudes as high as 5300 metres, flying both in daylight, night time, and during an almost complete solar eclipse.

Hess showed that the amount of ionising radiation decreased as one moved from ground level up to about 1000 metres, but then increased again rapidly. At 5300 metres, there was approximately twice as much ionising radiation as at ground level.[1] And because the effect occurred at night, and during a solar eclipse, it wasn’t due to the sun. Hess had proven that there was a source of this radiation outside the Earth’s atmosphere. Further unmanned balloon flights as high as 9 km showed the radiation increased even higher with altitude.

Readings of ionising radiation level (columns 2 to 4) at different altitudes (column 1, in metres), as recorded by Victor Hess. (Figure reproduced from [1].)

What this mysterious radiation was remained unknown until the late 1920s. It was initially thought to be electromagnetic radiation (i.e. gamma rays and x-rays). Robert Millikan named them cosmic rays in 1925 after proving that they originated outside the Earth. Then in 1927 the Dutch physicist Jacob Clay performed measurements while sailing from Java to the Netherlands, which showed that their intensity increased as one moved from the tropics to mid-latitudes.[2] He correctly deduced that the intensity was affected by the Earth’s magnetic field, which implied the cosmic rays must be charged particles.

Data recorded by Jacob Clay showing change in ionising radiation with latitude during his voyage from Java to Europe. (Figure reproduced from [2].)

In 1930, the Italian Bruno Rossi realised that if cosmic rays are electrically charged, then they should be deflected either east or west by the Earth’s magnetic field, depending on whether they are positively or negatively charged, respectively.[3] Experiments found that at all locations on the Earth’s surface there are more cosmic rays coming from the west than from the east, showing that most (if not all) cosmic ray particles are positively charged. This observation was called the east-west effect.

Illustration of the east-west effect. In the space around the Earth (shown as black in this diagram), the Earth’s magnetic field is directed perpendicularly out of the diagram. Incoming cosmic rays are shown in red. When they encounter Earth’s magnetic field, charged particles are deflected perpendicular to the field direction. Positively charged particles are deflected to the right, as shown, meaning that from the surface of the Earth, cosmic rays tend to preferentially come from the west.

Subsequent experiments determined that around 90% of cosmic rays entering our atmosphere are protons, 9% are helium nuclei (or alpha particles), and the remaining 1% are nuclei of heavier elements, with an extremely small number of other types of particles. And in 1936, for his crucial part in the discovery in cosmic rays, Victor Hess was awarded the Nobel Prize for Physics.

The origin of cosmic rays is however still not entirely clear. Our sun produces energetic particles that reach Earth, but cosmic rays are generally defined as coming from outside our own solar system. Our Milky Way Galaxy produces some of the lower energy particles, mostly from the direction of the galactic core, however at very high energies there is a deficit of cosmic rays in that direction, implying a shadowing effect on rays whose origin lies outside our galaxy. Known sources of cosmic rays include supernova explosions, supernova remnants (such as the Crab Nebula), active galactic nuclei, and quasars. But there are some very high energy cosmic rays whose source is still a mystery.

The fact that high energy cosmic rays originate from outside our galaxy, means that they should be isotropic – uniform in intensity distribution, independent of the direction from which they approach Earth. However, the Earth is not a stationary observation platform. The Earth orbits the sun at a speed of almost 30 km/s. But on a galactic scale this motion is dwarfed by the sun’s orbital speed around the core of our galaxy, which is 230 km/s, roughly in the direction of the star Vega, in the constellation Lyra. So relative to extragalactic cosmic rays, Earth is moving at an average speed of approximately 230 km/s. This speed adds to the energy of cosmic rays coming from the direction of Vega, and subtracts from the energies of cosmic rays coming from the opposite direction.

This difference is known as the Compton-Getting effect after the discoverers Arthur Compton and Ivan Getting.[4] It produces about a 0.1% difference in the energies of cosmic rays coming from the opposite directions, which can be observed statistically. The effect was confirmed experimentally in 1986.[5]

So we have two different observational effects that have been experimentally confirmed in the distribution of cosmic rays arriving at Earth. The Compton-Getting effect shows that the Earth is moving in the direction of the star Vega. Vega is of course above the Earth’s horizon as seen from half the planet’s surface at any one time, and below the horizon (behind the planet) from the other half of the Earth’s surface. By measuring cosmic ray distributions, you can show that the direction defined by the Compton-Getting anisotropy relative to the ground plane varies depending on your position on Earth. In other words, by measuring cosmic rays, you can prove that the Earth’s direction of motion through the galaxy is upwards from the ground in one place, while simultaneously downwards into the ground from a point on the opposite side of the planet, and at intermediate angles in places in between. Which is perfectly consistent for a spherical planet, but inconsistent with a Flat Earth.

The second effect, the east-west effect, is also readily explained with a spherical Earth, with the addition of a simple dipole magnetic field. As can be seen in the diagram above (“Illustration of the east-west effect”), incoming positively charged cosmic rays are uniformly deflected to the right (as viewed from above Earth’s North Pole), resulting in more rays arriving from the west than from the east, independent of location or time of day. The same observed east-west effect could in theory be produced on a Flat Earth, but only if the magnetic field is flattened out as well, holding the same relative orientation to the Earth’s surface as it does on the globe.

Shape of magnetic fields to produce the observed east-west effect in incoming cosmic rays. The required magnetic field for a spherical Earth is very close to a simple dipole, easily generated with known physical principles. The required magnetic field shape for a flat Earth is severely flattened, and cannot be produced with a simple magnetic dynamo model.

This would result in the field being grossly distorted from that of a simple dipole, and thus requiring some exotic method of generating such a complex field – a complex field that just happens to mimic exactly the field of a straightforward dipole if the Earth were spherical. In another application of Occam’s razor (similar to its use in article 8. Earth’s magnetic field), it is more parsimonious to conclude that the Earth is not flat, but spherical.

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What could be more simple than dropping an object and watching it fall to the ground? Our everyday experience shows that if you drop something, it falls straight down.

The Ancient Greek philosopher Aristotle used this observation to show that the Earth cannot possibly be moving or rotating. If the Earth were rotating from west to east, as some rival philosophers argued, then when you drop something, the Earth will move eastwards underneath it as it falls, and the object should land some distance west of where you dropped it! We don’t see this happening, ergo, the Earth cannot possibly be moving. Q.E.D.

Aristotle of course got many things wrong in his proposed system of the mechanics of motion and how the cosmos worked – a system known now as Aristotelian physics. Among his other contentions were that heavier objects fall faster than lighter ones, and that an object cannot undergo “unnatural motion” unless acted on by a force (falling is a “natural motion” and therefore requires no force).

One consequence of strict Aristotelian physics is that as soon as an object is released, it can have no sideways motion, and must fall straight down. This is obviously false if you observe objects such as arrows or cannonballs in flight, and natural philosophers of the Middle Ages developed the concept of impetus to explain this. The basic idea is that when a force propels an object, it implants in that object an impetus, which acts as an inherent force within the object itself, pushing it onwards. The concept took several centuries to mature, and was formalised by the French philosopher Jean Buridan in the 14th century (perhaps more famous for Buridan’s ass, which was not his backside, but a philosophical paradox).

Impetus is a forerunner of how Isaac Newton eventually solved the problem, with his idea of momentum and his Laws of Motion. Newton’s First Law says that an object at rest, or in a state of uniform motion, maintains that state unless acted on by an external force. So an object like an arrow that is fired from a bow at some speed maintains that speed unless acted on by external forces. In practice there are two forces acting on a flying arrow: air resistance, which slows down its horizontal motion, and gravity (another of Newton’s cool ideas), which causes it to begin falling towards the ground. The combination of these two results in the arrow slowing down and dropping in altitude, until eventually it hits something (either a target or the ground).

Newton’s First Law also explains why a dropped object falls straight to the ground, instead of falling to the west as the Earth rotates beneath it. An object held in your hand has the same rotational velocity as the Earth at the point where you’re standing. For example, if you’re on the equator, the rotational speed of the Earth’s surface is about 460 metres per second, so you and anything you’re holding, are moving eastwards at that speed. When you let the object go, it continues moving at 460 m/s eastward as it falls – the same speed as the Earth is moving. And so it falls at your feet, what appears to be directly downwards to you, with no sideways deflection.

If the Earth were flat and non-rotating, then none of this would be an issue. Objects dropped fall downwards and the non-motion of the Earth doesn’t change anything. You don’t even need Newton’s First Law.

If the Earth were flat and rotating like a record (or optical disc) about a central North Pole, things get a little more complicated. Let’s say the equator is 10,000 km from the North Pole (the same as on our Earth), and the Flat Earth rotates once per 24 hours. Then points on the equator are moving with a speed of 2π×10000/(24×60×60) km/s = 727 m/s (faster than our spherical Earth because the geometry is different). If you drop an object (and believe Newton’s laws), the object is moving east at 727 m/s and maintains that motion in a straight line as it falls. Let’s say you drop it from a height of 2 metres. It takes 0.64 seconds to hit the ground. In 0.64 seconds, the object moves east a distance of 465 metres. The ground is also moving east at 727 m/s, however, the ground is not moving in a straight line – it’s moving in a circle about the North Pole. In 0.64 seconds the ground moves through an angle of 0.0027°. Doing the trigonometry, this means the ground moves 465×cos(0.0027°) m east (which is slightly less than, but so close to 465 m that it’s not worth writing the difference) and 465×sin(0.0027°) m north, which equals 0.022 metres. So, if we live on a rotating Flat Earth, and you drop an object from 2 metres height at the equator, you should see it land 22 millimetres south of straight down.

On a rotating Flat Earth, if you drop an object on the equator (green dot, left), both the Earth and the object at that point are moving west to east. As the object falls, the Earth rotates (right). The location on the Earth where you dropped the object moves in a circle (green dot), but the falling object moves in a straight line and lands at the red dot, south of the starting location.

This is a prediction of the rotating Flat Earth model. Repeating the above calculations for different latitudes (assuming distances from the North Pole equal to our round Earth), we would expect a southward deflection of 11 mm at 45° north, or 33 mm at 45° south. Do we observe such a southward deflection of falling objects? No, we don’t.

Now let’s think about our spherical Earth, because things are not quite as straightforward as they might appear. The equator is rotating at a speed of 460 m/s. It’s not moving in a straight line – it’s rotating about the Earth’s axis, once every sidereal day: T = 23 hours, 56 minutes, 4 seconds (see 36. The visible stars). The radius of rotation is the equatorial radius of the Earth: 6378.1 km. If you hold an object 2 metres above the ground, its radius of rotation is 2 metres larger, making the distance it has to travel in a sidereal day 4π metres larger. So the speed of rotation 2 metres above the ground is 4π/T = 0.000145 m/s faster than the ground. In 0.46 seconds, this means that an object 2 metres above the ground moves eastward by 0.067 millimetres greater distance than the ground does.

So if you stand on the equator and drop an object from 2 metres, it should land 0.067 mm east of straight down. If you increase the height and drop an object from 100 metres, it takes 4.52 seconds to fall and by this calculation should land 33 mm east of straight down. The German language Wikipedia has an article on this calculation (there is no English Wikipedia article on it), and derives the same result, but then it says that “a more precise calculation” produces an additional factor 2/3, citing Carl Friedrich Gauss’s collected works without any further explanation. So this gives a deflection of 22 mm for an object dropped from 100 m. There is also an adjustment for latitude, being the usual cosine(latitude) term that we have seen in many of these discussions.

This is a prediction of the rotating spherical Earth model. Do we observe such an eastward deflection of falling objects?

There is in fact a long history of scientists investigating this effect and trying to measure it. In 1674, the French Jesuit priest and mathematician Claude François Milliet Dechales published his Cursus seu Mondus Matematicus, which included a diagram showing the fall of an object from a tower on the rotating Earth. It’s not clear if he ever performed the experiment.

Diagram from Dechales’s Cursus seu Mondus Matematicus showing an object F falling from a tower FG. As the Earth rotates the tower FG moves to HI, but the object does not land at I, it lands further east at L. (Public domain image from Wikimedia Commons.)

Isaac Newton himself wrote about the effect in a letter to Robert Hooke, dated 28 November, 1679, just five years later[2].

Newton’s letter of 28 November 1679 to Robert Hooke. Larger version. (Pages reproduced from [2].)

In the letter, Newton drew a diagram of an object falling, not just from a height to the ground, but continuing to fall towards the centre of the Earth (as if the object could pass through the Earth):

Enlargement of the diagram from Newton’s letter.

In the text accompanying the diagram Newton writes:

Then imagine this body be let fall and its gravity will give it a new motion towards the centre of the Earth without diminishing the previous one from west to east. Whence the motion of this body from west to east, by respect that before its fall it was more distant from the centre of the Earth than the parts of the Earth at which it arrives in its fall, will be greater than the motion from west to east of parts of the Earth at which the body arrives in its fall, and therefore it will not descend the perpendicular AC, but outrunning the parts of the Earth will shoot forward to the east side of the perpendicular, describing in its fall a spiral line ADEC.

Newton goes on to suggest that a “descent of but 20 to 30 yards” may be enough to observe the eastward deflection. Being a theoretician, Newton doesn’t seem to have done the experiment, but Hooke tried to measure the eastward deflection of an object falling from a height of 8.2 metres. From this height the expected deflection is about a quarter of a millimetre at the latitude of London—very difficult to measure—and Hooke’s results were inconclusive.

The first positive result was achieved in 1791 by Italian scientist Giovanni Battista Guglielmini. He dropped a total of 16 balls from the top of the Asinelli Tower of Bologna, a height of 78 m, comparing the landing positions to a vertical defined by a plumb-bob line. He concluded that the average eastward deflection of the balls was about 18 mm, compared to a predicted deflection of 11 mm.[3] Of course, these early experiments faced many difficulties, such as air currents, the difficulty of releasing the balls without any sideways motion, and measuring a vertical plumb line accurately.

In 1802, Johann Benzenberg dropped 32 balls from the tower of St Michael’s Church in Hamburg, 76 m high. Being at a higher latitude than Bologna, the expected eastward deflection was 8.7 mm, and Benzenberg recorded an average value of 9 mm. In 1831, Ferdinand Reich dropped lead balls 158 metres down the Drei Brüders (Three Brothers) mine shaft near Freiberg, measuring 28 mm eastward deflection, with a predicted value of 29.4 mm. In 1902 Edwin Hall performed the experiment with 948 separate drops from a height of 23 m at Harvard University, measuring an eastward deflection of 1.5 mm, compared to the predicted 1.8 mm. And Camille Flammarion dropped 144 balls from 68 m in the Pantheon in Paris, measuring a deflection of 6.3 mm, compared to the theoretical 8.1 mm.[3]

This is not an easy experiment to perform with sufficient accuracy. It is sensitive to a lot of complicating factors, particularly air currents, but the overall agreement of observation with the predictions is good. And so the measurable eastward deflection of falling objects provides us with another proof that the Earth is a (rotating) globe.

Note: I’ve talked only about the eastward deflection of falling objects. There is also a smaller predicted deflection in the north-south direction for latitudes away from the equator. That will be discussed in a future Proof.

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Our planet is made largely of rocks and metals. The composition and physical state varies with depth from the core of the Earth to the surface, because of changes in pressure and temperature with depth. The uppermost layer is the crust, which consists of lighter rocks in a solid state. Immediately below this is the upper mantle, in which the rocks are hotter and can deform plasticly over millions of years.

Slow convection currents occur in the upper mantle, and the convection cells define the tectonic plates of the Earth’s crust. Where mantle material rises, magma can emerge at mid-oceanic ridges or volcanoes. Where it sinks, a subduction zone occurs in the crust.

The plate boundaries are thus particularly unstable places on the Earth. As the plates shift and move relative to one another, stresses build up in the rock along the edges. At some point the stress becomes too great for the rock to withstand, and it gives way suddenly, releasing energy that shakes the Earth locally. These are earthquakes.

The point of slippage and the release of energy is known as the hypocentre of the earthquake, and may be several kilometres deep underground. The point on the surface above the hypocentre is the epicentre, and is where potential destruction is the greatest. Most earthquakes are small and go relatively unnoticed except by the seismologists who study earthquakes. Sometimes a quake is large and can cause damage to structures, injuries, and loss of life.

The energy released in an earthquake travels through the Earth in the form of waves, known as seismic waves. There are a few different types of seismic wave.

Primary waves, or P waves, are compressional waves, like sound waves in air. The rock alternately compresses and experiences tension, in a direction along the axis of propagation. In fact P waves are essentially sound waves of very large amplitude, and they propagate at the speed of sound in the medium. Within surface rock, this is about 5000 metres per second. Primary waves are so called because they are the fastest seismic waves, and thus the first ones to reach seismic recording stations located at any distance from the epicentre. They travel through the body of the Earth. And like sound waves, they can travel through any medium: solid, liquid, or gas.

Secondary waves, or S waves, are transverse waves, like light waves, or waves travelling along a jiggled rope. The rock jiggles from side to side as the wave propagates perpendicular to the jiggling motions. S waves travel a little over half the speed of P waves, and are the second waves to be detected at remote seismic stations. S waves also travel through the body of the Earth, but only within solid material. Fluids have no shear strength, and so cannot return to an equilibrium position when a transverse wave hits it, so the energy is dissipated within the fluid.

Illustrations of rock movement in different types of seismic waves. (Figure reproduced from [1].)

Besides these two types of body waves, there are also surface waves, which travel along the surface of the Earth. One type, Rayleigh waves (or R waves, named after the physicist Lord Rayleigh), are just like the surface waves or ripples on water, and causes the surface of the Earth to heave up and down. Another type of surface wave causes side to side motion; these are known as Love waves (or L waves, named after the mathematician Augustus Edward Hough Love). These waves propagate more slowly than S waves, at around 90% of the speed. Love waves are generally the strongest and most destructive seismic waves.

The P and S waves are thus the first two waves detected from an earthquake, and they are easily distinguishable on seismometer recordings.

Seismogram recording of arrival of P waves and S waves at a seismology station in Mongolia, from an earthquake 307 km away. (Figure reproduced from [2].)

The P waves arrive first and produce a pulse of activity which slowly fades in amplitude, then the S waves arrive and cause a larger amplitude burst of activity. Because the relative speeds of the two waves through the same material are known, the time between the arrival of the P and S waves can be used to determine the distance from the seismic station to the earthquake hypocentre, using a graph such as the following:

The graph shows the travel times of P, S, and also L waves, plotted against distance from the earthquake on the vertical axis. As you can see, the time between the detection of the P and S waves increases steadily with the distance from the quake.

Of course, if you have more than three seismic stations, you can pinpoint the location of the earthquake much more reliably and precisely. According to the International Registry of Seismograph Stations, there are over 26,000 seismic stations around the world.

Location of seismic stations recorded in the International Registry of Seismograph Stations. (Figure reproduced from [3].)

Interestingly, notice how the world’s seismic stations are concentrated along plate boundaries, where earthquakes are most common, particularly around the Pacific rim, as well as heavily in the developed nations of the US and Europe.

As shown in the travel-time curve graph, you can also use the propagation time of L waves to estimate distance to the earthquake. Did you notice the difference between the shapes of the P and S wave curves, and the L wave curve? L waves travel along the surface of the Earth. The distance from an earthquake to a detection station is measured conventionally, like everyday distances, also along the surface of the Earth. Since the L waves propagate at a constant speed, the graph of distance (along the Earth’s surface) versus time is a straight line.

But the P and S waves don’t travel along the surface of the Earth. They propagate through the bulk of the Earth. The distance that a P or S wave needs to travel from earthquake to detection site increases more slowly than the distance along the surface of the Earth, because of the Earth’s spherical shape. The S waves are only about 10% faster than the L waves, and you can see that near the epicentre, they arrive only around 10% earlier than the L waves. But the further away the earthquake is, the more of a shortcut they can take through the Earth, and so the faster they arrive, resulting in the downward curve on the graph. Similarly for the P waves.

This is in fact not the only cause of the P and S waves appearing to get faster the further away you are from an earthquake. They actually do get faster as they travel deeper, because of changes to the rock pressure. Deep in the Earth they can travel at roughly twice the speed that they do near the surface. The combination of these effects causes the shape of the curves in the travel-time graph.

If we consider the propagation of seismic waves from an earthquake, they spread out in circles around the epicentre, like ripples in a pond from where a stone is dropped in. The arrival times of the waves at seismic stations equidistant from the epicentre should be the same, since the speeds in any direction are the same. And this is of course what is observed. The following figures show the predicted spread of P waves across the Earth from earthquake epicentres in Washington State USA, near Panama, and near Ecuador, as plotted by the US Geological Survey.

Predicted P wave propagation time in minutes from an earthquake epicentre in Washington State, USA. (Public domain image from United States Geological Survey.)

Predicted P wave propagation time in minutes from an earthquake epicentre near Panama. (Public domain image from United States Geological Survey.)

Predicted P wave propagation time in minutes from an earthquake epicentre near Ecuador. (Public domain image from United States Geological Survey.)

These maps are shown on an equirectangular map projection, which of course distorts the shape of the surface of the Earth (as discussed in 14: Map projections). To get a better idea of how the seismic waves propagate, we need to project these maps onto a sphere.

Predicted P wave propagation time in minutes from an earthquake epicentre in Washington State, USA, projected onto a globe.

Predicted P wave propagation time in minutes from an earthquake epicentre near Panama, projected onto a globe.

Predicted P wave propagation time in minutes from an earthquake epicentre near Ecuador, projected onto a globe.

In these projections, you can see that the seismic wave travel time isochrones are circles, spreading out around the globe from the epicentres.

At least, the waves spread out in circles on a spherical Earth. In a flat Earth model, such as the typical “north pole in the middle” one, the spread of seismic waves produces elongated elliptical shapes or kidney shapes (such as the ones drawn in 23: Straight line travel), for no apparent or explicable reason.

Predicted P wave propagation time in minutes from an earthquake epicentre in Washington State, USA, projected onto a flat Earth.

Predicted P wave propagation time in minutes from an earthquake epicentre near Panama, projected onto a flat Earth.

Predicted P wave propagation time in minutes from an earthquake epicentre near Ecuador, projected onto a flat Earth.

Why should seismic waves propagate more slowly towards or away from the North Pole, and faster along tangential arcs? Why would they take longer to reach an area in the middle of the opposing half of the disc than to reach the far edge of the disc, which is further away? There is no a priori reason, and any proposed justification is yet another ad hoc bandage on the model.

So the propagation speeds of the various seismic waves and the travel times to recording stations provide another proof that the Earth is a globe.

Note: There is more to be said about the propagation of seismic waves, which will provide another, different proof that the Earth is a globe. Some readers no doubt have a good idea what it is already. Rest assured that I haven’t overlooked it, and it will be covered in detail in a future article.

This graph shows temperature anomalies on Earth – that is, the difference between the recorded temperature on any given day and the average temperature for the same location on that day over many years. Yellow-red colours indicate the actual temperature was warmer than average, blue-green colours indicate the temperature was cooler than average. The results are averaged across latitudes, so each point on the graph shows the average anomaly for the entire circle of latitude. The data are Goddard Television Infrared Observation Satellite Operational Vertical Sounder surface air temperature readings from NOAA polar weather satellites.

As you might expect, the temperature across Earth varies a bit. Some days are a bit warmer than average and some a bit cooler than average. You might imagine that with all of the different effects that go into the complicated atmospherical systems that control our weather, days would be cooler or warmer than average pretty much at random.

However that’s not what we’re seeing here. There’s a pattern to the anomalies. Firstly, the anomalies in the polar regions are larger (red and dark blue) than the anomalies in the mid-latitudes and tropic (yellow and light blue). Secondly, there are hints of almost regular vertical stripes in the graph – alternating bands of yellow and blue in the middle, and alternating red and dark blue near the poles. If you look at the graph carefully, you may be able to pick out a pattern of higher and lower temperatures, with a period a little bit less than one month.

What could have an effect on the Earth’s climate with a period a little under a month? The answer is, somewhat astonishingly, the moon.

The creators of this graph took the latitude-averaged temperature anomaly data for the 20 years from 1979 to 1998, and plotted it as a function of the phase of the moon:

These graphs show that the temperature anomalies have a clear relationship to the phase of the moon. In the polar regions, the temperature anomaly is strongly positive around the full moon, and negative around the new moon. In the mid-latitudes and tropics the trend is not so strong, but the anomalies tend to be lower around the full moon and positive around the new moon – the opposite of the polar regions.

What on Earth is going on here?

Aggregated measurements show that the polar latitudes of Earth are systematically around 0.55 degrees Celsius warmer at the full moon than at the new moon. This effect is strong enough that it dominates over the weaker reverse effect of the mid-latitudes/tropics anomaly. The average temperature of the Earth across all latitudes is not constant – it varies with the phase of the moon, dominated by the polar anomalies, being 0.02 degrees Celsius warmer at the full moon than the new moon. That doesn’t sound like a lot, but the signal is consistently there over all sub-periods in the 20-year data, and it is highly statistically significant.

The next puzzle is: What could possibly cause the Earth’s average temperature to vary with the phase of the moon?

Well, the full moon is bright, whereas the new moon is dark. Could the moonlight be warming the Earth measurably? Physicist and climate scientist Robert S. Knox has done the calculations. It turns out that the additional visible and thermal radiation the Earth receives from the full moon is only enough to warm the Earth by 0.0007 degrees Celsius, nowhere near enough to account for the observed difference[2].

There’s another effect of the moon’s regular orbit around the Earth. According to Newton’s law of gravity, strictly speaking the moon does not move in an orbit around the centre of the Earth. Two massive bodies in an orbital relationship actually each orbit around the centre of mass of the system, known as the barycentre. When one body is much more massive than the other, for example an artificial satellite orbiting the Earth, the motion of the larger body is very small. But our moon is over 1% of the mass of the Earth, so the barycentre of the system is over 1% of the distance from the centre of the Earth to the centre of the moon.

It turns out the Earth-moon barycentre is 4670 km from the centre of the Earth. This is still inside the Earth, but almost 3/4 of the way to the surface.

Animation showing the relative positions of the Earth and moon during the lunar orbital cycle. The red cross is the barycentre of the Earth-moon system, and both bodies orbit around it. Diagram is not to scale: relative to the Earth the moon is actually a bit larger than that (1/4 the diameter), and much further away (30× the Earth’s diameter). (Public domain image from Wikimedia Commons.)

The result of this is that during a full moon, when the moon is farthest from the sun, the Earth is 4670 km closer to the sun than average, whereas during a new moon the Earth is 4670 km further away from the sun than average. The Earth oscillates over 9000 km towards and away from the sun every month. And the increase in incident radiation from the sun during the phases around the full moon comes to about 43 mW per square metre, or an extra 5450 GW over the entire Earth. The Earth normally receives nearly 44 million GW of solar radiation, so the difference is relatively small, but it’s enough to heat the Earth by almost 0.01 degrees Celsius, which is near the observed average monthly temperature variation.

Why are the polar regions so strongly affected by this lunar cycle, while the tropics are weakly affected, and even show an opposing trend? Earth’s weather systems are complex and involve transport of heat across the globe by moving air masses. The burst of heat at the poles during a full moon actually migrates towards lower latitudes over several days – you can see the trend in the slope of the warm parts of the graph. The exact details of the physical mechanisms for these observations are still under discussion by the experts. What is clear though is that there is a definite cycle in the Earth’s average temperature with a period equal to the orbit of the moon, and it is most likely driven by the fact that the Earth is closer to the sun during a full moon.

How might one possibly explain this in a flat Earth model? Well, the “orbital” mechanics are completely different. The phase of the moon should have no effect on the distance of the Earth to the sun. The only moderately sensible idea might be that the full moon emits enough extra radiation to warm up the Earth. But the observations of the moon’s radiant energy and the amount of heating it can supply end up the same as the round Earth case (if you believe the same laws of thermodynamics). The full moon simply doesn’t supply anywhere near enough extra heat to the flat Earth to account for the observations.

One could posit that the sun varies in altitude above the flat Earth, coincidentally with the same period as the moon, thus providing additional heating during the full moon. However one of the main modifications to the geometry of the Earth-sun system made in flat Earth models is to fix the sun at a given distance (usually a few thousand kilometres) above the surface of the Earth, in an attempt to explain various geometrical properties such as the angle of the sun as seen from different latitudes. Letting the sun move up and down would mess up the geometry, and should easily be observable from the surface of the flat Earth.

So, observations of the global average temperature, and its periodic variation with the phase of the moon provides another proof that the Earth is a globe.

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The earliest method of marking time during the day was by following the movements of the sun as it crossed the sky, from sunrise in the east to sunset in the west. The apparent motion of the sun makes the shadows of fixed objects move during the day too. If you poke a stick into the ground, the shadow of the stick moves across the ground as time passes. By making marks on the ground and seeing which one the shadow is near, you get a method of telling the time of day. This is a simple form of sundial.

The apparent motion of the sun in the sky is caused by the interaction between the Earth’s orbit around the sun and the rotation of the Earth on its axis, which is inclined at approximately 23.5° to the axis of the orbital plane. At the June solstice (roughly 21 June), the northern hemisphere is maximally pointed towards the sun, making it summer while the southern hemisphere has winter. Half a year later at the December solstice, the sun is on the other side of the Earth, making it summer in the south and winter in the north. Midway between the solstices, at the March and September equinoxes, both hemispheres receive the same amount of sun.

Diagram of the interaction between Earth’s orbit and its tilted axis of rotation, showing the solstices and equinoxes that generate the seasons.

From the point of view of an observer standing on the Earth’s surface, the motions of the Earth make it appear as though the sun moves across the sky once per day, and drifts slowly north and south throughout the year. The following diagram shows the path of the sun across the sky for different dates, for my home of Sydney (latitude 34°S).

Sun’s path across the sky for different dates at latitude 34°S. (Diagram produced using [1].)

In the diagram, the horizon is around the edge, and the centre of the circles is directly overhead. The blue lines show the sun’s path for the indicated dates of the year. The sun is lowest in the sky to the north, and visible for the shortest time, on the June solstice (the southern winter), while it is highest in the sky and visible for the longest on the December solstice (in summer). The red lines show the position of the sun along each arc at the labelled hour of the day. For a location in the northern hemisphere north of the tropics, the sun paths would be curved the other way, passing south of overhead. In the tropics (between the Tropics of Capricorn and Cancer), some paths are to the north while some are to the south. On the equinoxes (20 March and 21 September), the sun rises due east at 06:00 and sets due west at 18:00 – this is true for every latitude.

If you have a fixed object cast a shadow, that shadow moves throughout the course of a day. The next day, if the sun has moved north or south because of the slowly changing seasons, the path the shadow traces moves a little bit above or below the previous day’s path.

The ancient Babylonians and Egyptians used sundials, and the ancient Greeks used their knowledge of geometry to develop several different styles. Greek sundials typically used a point-like object, called the nodus, as the reference marker. The nodus could be the very tip of a stick, a small ball or disc supported by thin wires, or a small hole that lets a spot of sunlight through. The shadow of the nodus (or the spot of light in the case of a hole nodus) moves across a surface in a regular way, not just with time of day, but also with the day of the year. During the day, the point-like shadow of a nodus traces a path from west to east (as the sun moves east to west in the sky). Throughout the year, the daily path moves north and south as the sun moves further south or north in the sky due to the seasons.

A nodus-based sundial, on St. Mary’s Basilica, Kraków, Poland. The nodus is a small hole in the centre of the cross. The horizontal position of the spot of light in the centre of the cross’s shadow indicates a time of just after 1:45 pm; the vertical position indicates the date (as indicated by the astrological symbols on the sides). It could be either about 1/3 of the way into the sign of Gemini (about 31 May), or 2/3 of the way through Cancer (about 12 July). The EXIF data on the photo indicates it was taken on 16 July, so the nodus date is fairly accurate. This sundial is mounted on a vertical wall, not horizontally, so the shadow travels left to right in the northern hemisphere, rather than right to left as it does for a horizontal sundial. (Public domain image from Wikimedia Commons.)

There are two slight complications. The red lines in the sun’s path diagram show timing of the sun paths assuming the Earth’s orbit is perfectly circular, but in reality it is an ellipse, with the Earth nearest the sun in January and furthest away in June. Earth travels around that elliptical path at different speeds—due to Newton’s law of gravity and laws of motion—moving fastest at closest approach in January, and slowest in June. The result of this is that the daily interval between when the sun crosses the north-south line is 24 hours on average, but varies systematically through the year. This variation in the sun’s apparent motion has a period of one year.

The second complication occurs because of the tilt of the Earth’s axis to the ecliptic plane in which it orbits. The sun’s apparent movement in the sky is due west (parallel to the Earth’s equator) only at the equinoxes. On any other date it moves at an angle, with a component of motion north or south, as it moves up or down the sky with the seasons. This north-south motion is maximal at the solstices. So at the solstices the westward component of the sun’s motion is less than it is at the equinoxes, meaning that it appears to move westward across the sky more slowly (because part of its speed is being used to move north or south). This variation in the sun’s apparent motion has a period of half a year.

To get the total variation in the sun’s motion, we need to add these two components. Doing so gives us the equation of time. This is the amount of time by which the sun’s position varies from the ideal “circular orbit, non-inclined axial spin” case, as a function of the day of the year.

The equation of time (red), showing the two components that make it up: the component due to Earth’s elliptical orbit (blue dashed line) and the component caused by the Earth’s axial tilt (green dot-dash line). The total shows the number of minutes that the sun’s apparent motion is ahead of its average position.

What this means is that if you have a standard sort of simple sundial, the shadow moves at different speeds across the face on different dates of the year, resulting in the shadow getting a little bit ahead or a little bit behind clock time. To get the correct time as shown by a clock, you need to read the time off the sundial’s shadow and subtract the number of minutes given by the equation of time for that date.

But this is thinking about sundials with our modern mindest about how time works. We have decided to make the unit of time we call a “day” the average length of time that it takes the sun to return to its highest position in the sky, and then we’ve divided that day into 24 exactly equal hours. An hour on 20 March is exactly the same length as an hour on 21 June, or on 21 December. “Of course it is!” you say.

But it wasn’t always so. For most of history, a “day” was defined as either the time between one sunrise and the next, or one sunset and the next, or the time between when the sun was due south in the sky and when it returned to being due south again (in the northern hemisphere). Each of these definitions of a “day” vary in length throughout the year. Saudi Arabia officially used Arabic time up until 1968, which defined midnight (the start of a new day) to be at sunset each day, and clocks needed to be adjusted every day to track the shift in sunset through the seasons.

The definition of a day as the period between the sun being due south (or north) and returning to that position the next day, is called solar time. For most of human timekeeping history, this is what was used. The fact that some days were a bit longer or shorter than others was of no consequence when the sun itself was the best timekeeping tool that anyone had access to.

Our modern concept of an hour has its origins in ancient Egypt, around 2,500 BC. The Egyptians originally divided the night time period into 12 parts, marked by the rising of particular stars in the sky. Because the stars change with the seasons (as discussed in 36. The visible stars), they had tables of which stars marked which hours for different dates of the year. Because of precession of the Earth’s orbit, the stars fell out of synch with the tables over the course of several centuries.

The oldest non-sundial timekeeping device that still exists is a water clock dating from the reign of Amenhotep III, around 1350 BC. It was a conical bowl, which was filled with water at sunset, and had a small outflow drip hole that let water out at a roughly constant rate. Inside the bowl is a set of 12 level marks, showing the water level at each of the 12 divisions of the night. But not just one set of 12 marks – there are multiple sets of 12 markings, with different spacings, that show the passage of the night time hours for different months of the year, when the length of the night is different.

Ancient Egyptian water clock (not Amenhotep’s one mentioned in the text). Dating uncertain, but possibly a much later Roman-era piece (circa 30 BC). The lower panel shows an unrolled cast of the interior of the conical bowl, showing the 12 different vertical rows of 12 differently spaced holes, indicating variable length hours for different months of the year. (Figure reproduced from [2].)

The oldest sundial we have is also from ancient Egypt, dating from around 1500 BC, a piece of limestone with a hole bored in it for a stick, and shadow marks, 12 of them, for dividing the daylight hours into 12 parts.

So the ancient Egyptians were dividing both the daylight and night time parts of each day into 12 different-length parts for a total of 24 divisions. Through cultural contact, sundials became a common way to mark the 12 hours of daylight in many other Mediterranean and Middle Eastern civilisations too, including the ancient Greeks and Romans.

By the Middle Ages, Catholic Europe was still keeping time based on a division of daylight time into 12 variable-length hours, and this carried across to the canonical hours, marking the times of day for liturgical prayers:

Matins: the night time prayer, recited some time after midnight, but before dawn.

Lauds: the dawn prayer, taking place at first light.

Prime: recited during the first hour of daylight.

Terce: at the third hour of the day time.

Sext: at midday, at the sixth hour, when the sun is due south.

Nones: the ninth hour of the day time.

Vespers: the sunset prayer, at the twelfth hour of the daylight period.

Compline: the end of the working day prayer, just before bed time.

In the modern world we might interpret “the third hour” to be 9:00 am, halfway between 6:00 am and midday, but the canonical hours are guided by the sun, so Terce would be earlier in summer and later in winter, in the same way that sunrise, and hence the celebration of Lauds, are. Nones, in contrast, would be earlier in winter and later in summer. (Incidentally, we get our modern word “noon” from “Nones” – although you’ll notice that Nones was defined as the ninth hour, or around 3:00 pm. For some reason it moved to become associated with the middle of the day. We’re not sure exactly why, but historians believe that the monks who observed this liturgy fasted each day until after the prayer of Nones, so there was constant pressure to make it slightly earlier, which eventually moved it back a full three hours!)

You might think that when mechanical clocks were invented, people suddenly realised that they’d been doing things wrong the whole time, and they quickly moved to the modern system of an hour being of a constant length. But that’s not what happened. The first mechanical clocks used a verge escapement to regulate the motion of the gear wheels, and this remained the most accurate clock mechanism from the 13th century to the 17th. But it wasn’t very accurate, varying by around 15 minutes per day, and so verge clocks had to be reset daily to match the motion of the sun.

Christiaan Huygens invented the pendulum clock in 1656, vastly improving the accuracy of mechanical clocks, down to around 15 seconds per day. With this new level of accuracy, people fully realised for the first time that the length of a full day as measured by the time it took the sun to return to the highest position in the sky didn’t match a regularly ticking clock. But rather than adjust their definition of what an hour was, people decided there must be a way to get these regular clocks to tell proper solar time! Thus were invented equation clocks.

The first equation clocks had a correction dial, which essentially displayed the equation of time value for the current day of the year. You read the time off the main clock dial, and then added the correction displayed on the correction dial, and that gave you the “correct” solar time. By the 18th century, the correction gearing was incorporated into the main clock face display, so that the hands of the clock actually ran faster or slower at different times of the year, to match the movement of the sun. It wasn’t until the early 19th century that European society moved to a mean time system (“mean” as in “average”), in which each “day” was defined to be exactly the same length, and the hour was a fixed period of time (thus simplifying clockmakers’ lives considerably).

Just to complete this story, clocks in the early 19th century were set to local mean time, which was the mean time of their meridian of longitude. Towns a few tens of miles east or west would have different mean times by a few minutes. This caused problems beginning with the introduction of rapid travel enabled by the railways, eventually leading to the adoption of standard time zones in the 1880s, in which all locations in slices of roughly 1/24 of the Earth share the same time.

What this means is that people were still living their lives by local solar time up until the early 19th century. In other words, a sundial was still the most accurate method of telling the time up until just 200 years ago – and it didn’t need any corrections based on the equation of time because people weren’t using mean time yet. It’s only in the past 200 years that we’ve had to correct a sundial to give what we consider to be the correct clock time.

So, back to sundials. Assuming we are happy with solar time (and can use the equation of time to correct to mean time if we wish), the main thing we need to contend with is that the sun moves north and south in the sky throughout the year. A nodus-type sundial accounts for this by marking lines that indicate the time when the shadow of the nodus crosses them on different days of the year. But many sundials use the whole edge of a stick or post as the shadow marker – this edge is called the gnomon. As the sun moves north and south throughout the year, different parts of the gnomon will cast their shadows in different places. If the gnomon is aligned parallel to the axis of the Earth, then these motions will be along the edge of the shadow, rather shifting the edge of the shadow laterally. You can then read solar time using a single marking, at any time of the year.

Another way to think about it is that from a viewpoint on Earth, the sun appears to revolve in the sky about the Earth’s axis. So if your sundial has a gnomon that is parallel to the Earth’s axis, the sun appears to rotate with the gnomon as its axis once per day, and the shadow of the gnomon indicates solar time on the marked surface below. As the sun moves north or south with the seasons, it is still revolving around the gnomon, so the shadow still tracks solar time accurately. If the gnomon is not parallel to the revolution axis, then as the sun moves north and south, the shadow of the gnomon will shift positions on the marked surface, and the time will be inaccurate at different times of the year.

This is why sundials with gnomons all have them inclined at an angle from the horizontal equal to the latitude of where the sundial is placed. At the North Pole, a vertical stick will indicate solar time accurately throughout the entire summer (when the sun is above the horizon 24 hours a day). At London (latitude 51.5°N), sundial gnomons are pointed north at 51.5° from the horizontal.

So, in order to work properly, gnomon-sundials must have a gnomon angled parallel to the Earth’s axis of rotation. The fact that sundials at different latitudes need to have their gnomons at different angles to the ground plane shows that the ground plane is only perpendicular to the Earth’s rotation axis at the North and South Poles, and the angle between the ground and Earth’s axis of rotation varies everywhere else in a way consistent with the Earth being a globe.

If the Earth were flat… well, all of this would just be a huge coincidence in the motion of the sun above the flat Earth, that for some unexplained reason exactly mimics the geometry of a spherical Earth in orbit about the sun. In fact, to get all of the angles to match sundial observations you need to posit that the sun’s rays don’t even travel in straight lines.

Addendum: I just wanted to show you this magnificent sundial, in the Monastery of Lluc, in Mallorca, Spain.

Top left shows the canonical hours. At sunrise (no matter what time sunrise happens to be), the shadow of the stick indicates the liturgy of Prime. Sext occurs at solar noon, when the sun is directly overhead, with Terce halfway between Prime and Sext. Vespers is at sunset (again, regardless of the modern clock time), with Nones halfway between Sext and Vespers. The night time hours of Complice, Matins, and Lauds are marked above the horizontal (and in fact would correctly indicate the times if the Earth were transparent, so the sun could cast a shadow from underneath the horizon).

Bottom left shows a nodus sundial, the tip of the stick marking “Babylonian” hours, which were used in Mallorca historically. This counts 0 (or 24) at sunrise, and then equal numbered hours thereafter. The vertical position of the nodus shadow marks the date (similar to the Krakow sundial above).

The central dial is a gnomon indicating “true solar time”. The shadow of the edge of the gnomon indicates the solar hour.

Finally the two dials on the right are nodus dials, showing mean time horizontally, and date of the year vertically. The top dial is to be read in summer and autumn, whole the lower dial is for winter and spring. It looks like the dials also include a daylight saving adjustment, assuming it begins and ends on the equinoxes!

The time (confirmed from the photo EXIF data) is 4:15 pm, and the date is 9 September, 12 days before the autumnal equinox (read on the top right dial).

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When our ancestors looked up into the night sky, they beheld the wonder of the stars. With our ubiquitous electrical lighting, many of us don’t see the same view today – our city skies are too bright from artificial light (previously discussed under Skyglow). We can see the brightest handful of stars, but most of us have forgotten how to navigate the night sky, recognising the constellations and other features such as the intricately structured band of the Milky Way and the Magellanic Clouds. There are features in the night sky other than stars (the moon, the planets, meteors, and comets), but we’re going to concentrate on the stars.

The Milky Way counts because it is made of stars. To our ancestors, it resembled a stream of milk flung across the night sky, a continuous band of brightness. But a small telescope reveals that it is made up of millions of faint stars, packed so closely that they blend together to our naked eyes. The Milky Way is our galaxy, a collection of roughly 100 billion stars and their planets.

The stars are apparently fixed in place with respect to one another. (Unlike the moon, planets, meteors, and comets, which move relative to the stars, thus distinguishing them.) The stars are not fixed in the sky relative to the Earth though. Each night, the stars wheel around in circles in the sky, moving over the hours as if stuck to the sky and the sky itself is rotating.

The stars move in their circles and come back to the same position in the sky approximately a day later. But not exactly a day later. The stars return to the same position after 23 hours, 56 minutes, and a little over 4 seconds, if you time it precisely. We measure our days by the sun, which appears to move through the sky in roughly the same way as the stars, but which moves more slowly, taking a full 24 hours (on average, over the course of a year) to return to the same position.

This difference is caused by the physical arrangement of the sun, Earth, and stars. Our Earth spins around on its axis once every 23 hours, 56 minutes, and 4 and a bit seconds. However in this time it has also moved in its orbit around the sun, by a distance of approximately one full orbit (which takes a year) divided by 365.24 (the average number of days in a year). This means that from the viewpoint of a person on Earth, the sun has moved a little bit relative to the stars, and it takes an extra (day/365.24) = 236 seconds for the Earth to rotate far enough for the sun to appear as though it has returned to the same position. This is why the solar day (the way we measure time with our clocks) is almost 4 minutes longer than the Earth’s rotation period (called the sidereal day, “sidereal” meaning “relative to the stars”).

Diagram showing the difference between a sidereal day (23 hours, 56 minutes, 4 seconds) when the Earth has rotated once, and a solar day (24 hours) when the sun appears in the same position to an observer on Earth.

Another way of looking at is that in one year the Earth spins on its axis 366.24 times, but in that same time the Earth has moved once around the sun, so only 365.24 solar days have passed. The sidereal day is thus 365.24/366.24 = 99.727% of the length of the solar day.

The consequence of all this is that slowly, throughout the year, the stars we see at night change. On 1 January, some stars are hidden directly behind the sun, and we can’t see them or nearby stars, because they are in the sky during the day, when their light is drowned out by the light of the sun. But six months later, the Earth is on the other side of its orbit, and those stars are now high in the sky at midnight and easily visible, whereas some of the stars that were visible in January are now in the sky at daytime and obscured.

This change in visibility of the stars over the course of a year applies mostly to stars above the equatorial regions. If we imagine the equator of the Earth extended directly upwards (a bit like the rings of Saturn) towards the stars, it defines a plane cutting the sky in half. This plane is called the celestial equator.

However the sun doesn’t move along this path. The Earth’s axis is tilted relative to its orbit by an angle of approximately 23.5°. So the sun’s apparent path through the sky moves up and down by ±23.5° over the course of a year, which is what causes our seasons. When the sun is higher in the sky it is summer, when it’s lower, it’s winter.

So as well as the celestial equator, there is another plane bisecting the sky, the plane that the sun appears to follow around the Earth – or equivalently, the plane of the Earth’s (and other planets’) orbit around the sun. This plane is called the ecliptic. It’s the stars along and close to the ecliptic that appear the closest to and thus the most obscured by the sun throughout the year.

The constellations of the ecliptic have another name: the zodiac. We’ve met this term before as part of the name of the zodiacal light. The zodiacal light occurs in the plane of the planetary orbits, the ecliptic, which is the same as the plane of the zodiac. As an aside, the constellations of the zodiac include those familiar to people through the pre-scientific tradition of Western astrology: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius (“Scorpio” in astrology), Ophiuchus (ignored in astrology), Sagittarius, Capricornus (“Capricorn” in astrology), Aquarius, and Pisces. The system of astrology abstracts these real-world constellations into 12 idealised segments of the sky, each covering exactly 30° of the circle (in fact the constellations cover different amounts), and assigns portentous meanings to the positions of the sun, moon, and planets within each segment.

The stars close to the zodiac are completely obscured by the sun for part of the year, while the stars near the celestial equator appear close to the sun but might still be visible (with difficulty) immediately after sunset or before dawn. The stars far from these planes, however, are more easily visible throughout the whole year. The north star, Polaris, is almost directly above the North Pole, and it and stars nearby are visible from most of the northern hemisphere year-round. There is no equivalent “south pole star”, but the most southerly constellations—such as the recognisable Crux, or Southern Cross—are similarly visible year-round through most of the southern hemisphere.

Diagram showing the axial tilt of the Earth relative to the plane of the orbit (the ecliptic), and the positions of Polaris and stars in the zodiac and on the celestial equator. Sizes and distances are not to scale – in reality Polaris is so far away that the angle it makes between the June and December positions of Earth is only 0.007 seconds of arc (about a five millionth of a degree).

Interestingly, Polaris is never visible from the southern hemisphere. Similarly, Crux is not visible from almost all of the northern hemisphere, except for a band close to the equator, from where it appears extremely low on the southern horizon. Crux is centred around 60° south, celestial latitude (usually known as declination), which means that it is below the horizon from all points north of latitude 30°N. (In practice, stars near the horizon are obscured by topography and the long path through the atmosphere, so it is difficult to spot Crux from anywhere north of about 20°N.)

In general, stars at a given declination can never be seen from Earth latitudes 90° or more away, and only with difficulty from 80°-90° away. The reason is straightforward enough. From our spherical Earth, if you are standing at latitude x°N, all parts of the sky from (90-x)°S declination to the south celestial pole are below the horizon. And similarly if you’re at x°S, all parts of the sky from (90-x)°N declination to the north celestial pole are below the horizon. The Earth itself is in the way.

On the other hand, if you are standing at latitude x°N, all parts of the sky north of the same declination are visible every night of the year, while stars between x°N and (90-x)°S are visible only at certain times of the year.

Visibility of stars from parts of Earth is determined simply by sightlines from the surface of the globe.

With a spherical Earth, the geometry of the visibility of stars is readily understandable. On a flat Earth, however, there’s no obvious reason why some stars would be visible from some parts of the Earth and not others, let alone the details of how the visibilities change with latitude and throughout the year.

If we consider the usual flat Earth model, with the North Pole at the centre of a disc, and southern regions around the rim, it is difficult to imagine how Polaris can be seen from regions north of the equator but not south of it. And it is even more difficult to justify how it is even possible for southern stars such as those in Crux being visible from Australia, southern Africa, and South America but not from anywhere near the centre of the disc. The southern stars can be seen in the night sky from any two of these locations simultaneously, but if you use a radio telescope during daylight you can observe the same stars from all three at once. Things get even worse with Antarctica. In the southern winter, it is night at virtually every location in Antarctica at the same time, and many of the same stars are visible, yet cannot be seen from the northern hemisphere.

Visibility of stars from a flat Earth. All stars must be above the plane, but why are some visible in some parts of the world but not others? Particularly the southern stars, which can be seen from widely separated locations but not regions in the middle of them.

In any flat Earth model, there should be a direct line of sight from every location to any object above the plane of the Earth. To attempt to explain why there isn’t requires special pleading to contrived circumstances such as otherwise undetectable objects blocking lines of sight, or light rays bending or being dimmed in ways inconsistent with known physics.

The fact that when you look up at night, you can’t see all the stars visible from other parts of the Earth, is a simple consequence of the fact that the Earth is a globe.

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In the opening years of the twentieth century, scientists in the field of geodesy (measuring the shape and gravitational field of the Earth) were interested in making measurements of the strength of gravity all over the Earth’s surface. To do this, they trekked to remote regions of the world with sensitive gravimeters, to take the readings. On land this was straightforward enough, but they also wanted measurements taken at sea.

Around 1900, teams from the Institute of Geodesy in Potsdam took voyages into the Atlantic, Indian, and Pacific Oceans on ships, and made measurements using their gravimeters. The collected data were brought back to Potsdam for analysis. There, the readings fell under the scrutinising eyes of the Hungarian physicist Loránd Eötvös, who specialised in studying the variation of Earth’s gravitational field with position on the surface. He noticed an odd thing about the readings.

Because of the impracticality of stopping the ship every time they wanted to take a reading, the scientists measured the Earth’s gravity while the ships were moving. There was no reason to suppose this would make any difference. But Eötvös found a systematic effect. Gravity measurements taken while the ship was moving eastward were lower than readings taken while the ship was moving westward.

Eötvös realised that this effect was being caused by the rotation of the Earth. The Earth’s equatorial circumference is 40,075 km, and it rotates eastward once every sidereal day (23 hours, 56 minutes). So the ground at the equator is rotating at a linear speed of 465 metres per second. To move in a circular path rather than a straight line (as dictated by Newton’s First Law of Motion), gravity supplies a centripetal force to any object on the Earth’s surface. The necessary force is equal to the object’s mass times the velocity squared, divided by the radius of the circular path (6378 km). This comes to m×4652/6378000 = 0.0339m. So per kilogram of mass, a force of 0.0339 newtons is needed to enforce the circular path, an amount easily supplied by the Earth’s gravity. (This is why objects don’t get flung off the Earth by its rotation, a complaint of some spherical Earth sceptics.)

What this means is that the effective acceleration due to gravity measured for an object sitting on the equator is reduced by 0.0339 m/s2 (the same units as 0.0339 N/kg) compared to if the Earth were not rotating. But if you’re on a ship travelling east at, say, 10 m/s, the centripetal force required to keep you on the Earth’s surface is greater, equal to 4752/6378000 = 0.0354 N/kg. This reduces the apparent measured gravity by a larger amount, making the measured value of gravity smaller. And if you’re on a ship travelling west at 10 m/s, the centripetal force is 4552/6378000 = 0.0324 N/kg, reducing the apparent gravity by a smaller amount and making the measured value of gravity greater. The difference in apparent gravity between the ships travelling east and west is 0.003 m/s2, which is about 0.03% of the acceleration due to gravity. For a person of mass 70 kg, this is a difference in apparent weight of about 20 grams (strictly speaking, a difference in weight of 0.2 newtons, which is 20 grams multiplied by acceleration due to gravity).

Eötvös set out these theoretical calculations, and then organised an expedition to measure and test his results. In 1908, the experiment was carried out on board a ship in the Black Sea, with two separate ships travelling east and west past one another so the measurements could be made at the same time. The results matched Eötvös’s predictions, thus confirming the effect.

In general (if you’re not at the equator), your linear speed caused by the rotation of the Earth is equal to 465 m/s times the cosine of your latitude, while the radius of your circular motion is also equal to 6378 km times the cosine of your latitude. The centripetal force formula uses the square of the velocity divided by the radius, so this results in a cosine(latitude) term in the final result. That is, the size of the Eötvös effect also varies as the cosine of the latitude. If you measure it at 60° latitude, either north or south, the difference in gravity between east and west travelling ships is half that measured at the equator.

The Eötvös effect is well known in the field of gravimetry, and is routinely corrected for when taking measurements of the Earth’s gravitational strength from moving ships[1], aircraft[2], or submarines[3]. The reference on submarines refers to a gravitational measurement module for use on military submarines to enhance their navigation capability as undersea instruments of warfare. This module includes an Eötvös effect correction for when the sub is moving east or west. You can bet your bottom dollar that no military force in the world would make such a correction to their navigation instruments if it weren’t necessary.

One paper I found reports measurements made of the detailed structure of gravitational anomalies over the Mariana Trough in the Pacific Ocean south of Japan. It states:

Shipboard free-air gravity anomalies were calculated by subtracting the normal gravity field data from observed gravity field data, with a correction applied for the Eötvös effect using Differential Global Positioning System (DGPS) data.[4]

The results look pretty cool:

Map of gravitational anomalies in the Mariana Trough region of the Pacific Ocean, as obtained by shipboard measurement, corrected for the Eötvös effect. (Figure reproduced from [4].)

Another paper shows the Eötvös effect more directly:

Graph showing measurements of Earth’s gravitational field strength versus distance travelled by a ship in the South Indian Ocean. In the leftmost section (16), the ship is moving slowly westward. In the central section (17) the ship is moving at a faster speed westward, showing the increase in measured gravity. In the right section (18) the ship is moving eastward at slow speed, and the gravity readings are lower than the readings taken in similar positions while moving westward. (Figure reproduced from [5].)

If the Earth were flat, on the other hand, there would be no Eötvös effect at all. If the flat Earth is not rotating (as most models posit, with the sun moving above it in a circular path), obviously there is no centripetal acceleration happening at all. Even if you adopt a model where the flat Earth rotates about the North Pole, the centripetal acceleration at every point on the surface is parallel to the surface, towards the pole, not directed downwards. So an Eötvös-like effect would actually cause a slight deflection in the angle of gravity, but almost zero change in the magnitude of the gravity.

The Eötvös effect shows that not only is the Earth rotating, but that it is rotating about a central point that is underneath the ground, not somewhere on the surface. If you stand on the equator and face east, the surface of the Earth is rotating in the direction you are facing and downwards, not to the left or right. Furthermore, the cosine term shows that at equal latitudes both north and south, the rotation is at the same angle relative to the surface, which can only be the case if the Earth is symmetrical about the equator: i.e. spherical.

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Opening disclaimer: I’m going to be talking about “heat” a lot in this one. Formally, “heat” is defined as a process of energy flow, and not as an amount of thermal energy in a body. However to people who aren’t experts in thermodynamics (i.e. nearly everyone), “heat” is commonly understood as an “amount of hotness” or “amount of thermal energy”. To avoid the linguistic awkwardness of using the five-syllable phrase “thermal energy” in every single instance, I’m just going to use this colloquial meaning of “heat”. Even some of the papers I cite use “heat” in this colloquial sense. I’ve already done it in the title, which to be technically correct should be the more awkward and less pithy “Earth’s internal thermal energy”.

The interior of the Earth is hot. Miners know first hand that as you go deeper into the Earth, the temperature increases. The deepest mine on Earth is the TauTona gold mine in South Africa, reaching 3.9 kilometres below sea level. At this depth, the rock temperature is 60°C, and considerable cooling technology is required to bring the air temperature down to a level where the miners can survive. The Kola Superdeep Borehole in Russia reached a depth of 12.2 km, where it found the temperature to be 180°C.

Deeper in the Earth, the temperature gets hot enough to melt rock. The results are visible in the lava that emerges from volcanic eruptions. How did the interior of the Earth get that hot? And exactly how hot is it down there?

For many years, geologists have been measuring the amount of thermal energy flowing out of the Earth, at thousands of measuring stations across the planet. A 2013 paper analyses some 38,374 heat flow measurements across the globe to produce a map of the mean heat flow out of the Earth, shown below[1]:

Mean heat flow out of the Earth in milliwatts per square metre, as a function of location. (Figure reproduced from [1].)

From the map, you can see that most of Earth’s heat emerges at the mid-ocean ridges, deep underwater. This makes sense, as this is where rising plumes of magma from deep within the mantle are acting to bring new rock material to the crust. The coolest areas are generally geologically stable regions in the middle of tectonic plates.

Although the heat flow out of the Earth’s surface is of the order of milliwatts per square metre, the surface has a lot of square metres. The overall heat flow out of the Earth comes to a total of around 47 terawatts[2]. In contrast, the sun emits close to 4×1014 terawatts of energy in total, and the solar energy falling on the Earth’s surface is 1360 watts per square metre, over 10,000 times as much as the heat energy leaking out of the Earth itself. So the sun dominates Earth’s heating and weather systems by roughly that factor.

So the Earth generates some 47 TW of thermal power. Where does this huge amount of energy come from? To answer that, we need to go all the way back to when the Earth was formed, some 4.5 billion years ago.

Our sun formed from the gaseous and dusty material distributed throughout the Galaxy. This material is not distributed evenly, and where there is a denser concentration, gravity acts to draw in more material. As the material is pulled in, any small motions are amplified into an overall rotation. The result is an accretion disc, with matter spiralling into a growing mass at the centre. When the central concentration accumulates enough mass, the pressure ignites nuclear reactions and a star is born. Some of the leftover material continues to orbit the new star and forms smaller accretions that eventually become planets or smaller bodies.

The process of accreting matter generates thermal energy. Gravitational potential energy reduces as matter pulls closer together, and the resulting collisions between matter particles convert it into thermal energy, heating up the accumulating mass. Our Earth was born hot. As the matter settled into a solid body, the shrinking further heated the core through the Kelvin-Helmholtz mechanism. The total heat energy from the initial formation of the Earth dissipates only very slowly into space, and that process is still going on today, 4.5 billion years later.

It’s not known precisely how much of this primordial heat is left in Earth or how much flows out, but various different studies suggest it is somewhere in the range of 12-30 TW, roughly a quarter to two-thirds of Earth’s total measured heat flux[3]. So that’s not the only source of the heat energy flowing out of the Earth.

The other source of Earth’s internal heat is radioactive decay. Some of the matter in the primordial gas and dust cloud that formed the sun and planets was produced in the supernova explosions of previous generations of stars. These explosions produce atoms of radioactively unstable isotopes. Many of these decay relatively rapidly and are essentially gone by now. But some isotopes have very long half-lives, most importantly: potassium-40 (1.25 billion years), thorium-232 (14.05 billion years), uranium-235 (703.8 million years), and uranium-238 (4.47 billion years). These isotopes still exist in significant quantities inside the Earth, where they continue to decay, releasing energy.

We have a way of probing how much radioactive energy is released inside the Earth. The decay reactions produce neutrinos (which we’ve met before), and because they travel unhindered through the Earth these can be detected by neutrino observatories. These geoneutrinos have energy ranges that distinguish them from cosmic neutrino sources, and of course always emerge from underground. The observed decay rates from geoneutrinos correspond to a total radiothermal energy production of 10-30 TW, of the same order as the primordial heat flux. (The neutrinos themselves also carry away part of the energy from the radioactive decays, roughly 5 TW, but this is an additional component not deposited as thermal energy inside the Earth.)

Approximate radiothermal energy generated within the Earth, plotted as a function of time, from the formation of the Earth 4.5 billion years ago, to the present. The four main isotopes are plotted separately, and the total is shown as the dashed line. (Public domain figure adapted from data in [4], from Wikimedia Commons.)

To within the uncertainties, the sum of the estimated primordial and measured radiothermal energy fluxes is equal to the total measured 47 TW flux. So that’s good.

Once you know how much heat is being generated inside the Earth, you can start to apply heat transfer equations, knowing the thermodynamic properties of rock and iron, how much conduction and convection can be expected, and cross-referencing it with our knowledge of the physical state of these materials under different temperature and pressure conditions. There’s also additional information about the internal structure of the Earth that we get from seismology, but that’s a story for a future article. Putting it all together, you end up with a linked series of equations which you can solve to determine the temperature profile of the Earth as a function of depth.

Temperature profile of the Earth’s interior, from the surface (left) to the centre of the core (right). Temperature units are not marked on the vertical axis, but the temperature of the surface (bottom left corner) is approximately 300 K, and the inner core (IC, right) is around 7000 K. UM is upper mantle, LM lower mantle, OC outer core. The calculated temperature profile is the solid line. The two solid dots are fixed points constrained by known phase transitions of rock and iron – the slopes of the curves between them are governed by the thermodynamic equations. The dashed lines are various components of the constraining equations. (Figure reproduced from [5].)

The results are all self-consistent, with observations such as the temperature of the rock in deep mine shafts and the rate of detection of geoneutrinos, with structural constraints provided by seismology, and with the temperature constraints and known modes of heat flow from the core to the surface of the Earth.

That is, they’re all consistent assuming the Earth is a spherical body of rock and iron. If the Earth were flat, the thermal transport equations would need to be changed to reflect the different geometry. As a first approximation, assume the flat Earth is relatively thin (i.e. a cylinder with the radius larger than the height). We still measure the same amount of heat flux emerging from the Earth’s surface, so the same amount of heat has to be either (a) generated inside it, or (b) being input from some external energy source underneath the flat Earth. However geoneutrino energy ranges indicate that they come from radioactive decay of Earthly minerals, so it makes sense to conclude that radiothermal heating is significant.

If radioactive decay is producing heat within the bulk of the flat Earth, then half of the produced neutrinos will emerge from the underside, and thus be undetectable. So the total heat production should be double that deduced from neutrino observations, or somewhere in the range 20-60 TW. To produce twice the energy, you need twice the mass of the Earth. If the flat Earth is a disc with radius 20,000 km (the distance from the North Pole to the South Pole), then to have the same volume as the spherical Earth it would need to be 859 km thick. But we need twice as much mass to produce the observed thermal energy flux, so it should be approximately 1720 km thick. Some fraction of the geoneutrinos will escape from the sides of the cylinder of this thickness, which means we need to add more rock to produce a bit more energy to compensate, so the final result will be a bit thicker.

There’s no obvious reason to suppose that a flat Earth can’t be a bit over 1700 km thick, as opposed to any other thickness. With over twice as much mass as our spherical Earth, the surface gravity of this thermodynamically correct flat Earth would be over 2 Gs (i.e. twice the gravity we experience), which is obviously wrong, but then many flat Earth models deny Newton’s law of gravity anyway (because it causes so many problems for the model).

But just as in the spherical Earth model the observed geoneutrino flux only accounts for roughly half the observed surface heat flux. The other half could potentially come from primordial heat left over from the flat Earth’s formation – although as we’ve already seen, what we know about planetary formation precludes the formation of a flat Earth in the first place. The other option is (b) that the missing half of the energy is coming from some source underneath the flat Earth, heating it like a hotplate. What this source of extra energy is is mysterious. No flat Earth model that I’ve seen addresses this problem, let alone proposes a solution.

What’s more, if such a source of energy under the flat Earth existed, then it would most likely also radiate into space around the edges of the flat Earth, and have observable effects on the objects in the sky above us. What we’re left with, if we trust the sciences of radioactive decay and thermal energy transfer, is a strong constraint on the thickness of the flat Earth, plus a mysterious unspecified energy source underneath – neither of which are mentioned in standard flat Earth models.